On the Cauchy problem in Sobolev spaces for nonlinear Schrodinger equations with potential

Preprint

23 January 2007

preprint
Published by arXiv in arXiv

https://doi.org/10.48550/arXiv.math/0609391

Abstract

We consider the Cauchy problem for nonlinear Schrodinger equations in the presence of a smooth, possibly unbounded, potential. No assumption is made on the sign of the potential. If the potential grows at most linearly at infinity, we construct solutions in Sobolev spaces (without weight), locally in time. Under some natural assumptions, we prove that the $H^1$-solutions are global in time. On the other hand, if the potential has a super-linear growth, then the Sobolev regularity of positive order is lost instantly, not matter how large it is, unless the initial datum decays sufficiently fast at infinity.

Keywords

NONLINEAR SCHRODINGER EQUATIONS
SCHRODINGER EQUATIONS
SOBOLEV SPACES
CAUCHY PROBLEM
INFINITY
HAND

All Related Versions

Version 1, 2006-09-14, ArXiv
Version 2, 2007-01-23, ArXiv
Version 1, 2014-09-19, ArXiv (Unconfirmed version)
Version 2, 2014-09-23, ArXiv (Unconfirmed version)
Version 3, 2014-10-01, ArXiv (Unconfirmed version)

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